What does the bookmark icon in the profile do?

The first icon on the page is to bookmark the profile. When you select the bookmark item on your own, or someone else's profile, the bookmark item will highlight to indicate that the profile has been bookmarked.

When you return to your timeline after bookmarking a profile and open the app drawer by tapping the three lines in the upper left, and select the Bookmarks item in the list, you will see that the profile now appears as a bookmark for quick access.